Constrained tuning: Difference between revisions

Line 11:

== Definition ==

Given a ~~temperament~~ [[mapping]] ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation, represented by transformation matrix ''X'', and a ''q''-norm. Suppose the tuning is constrained by the eigenmonzo list ''M''~~~~''I''~~. The goal is to find~~ the generator ~~list ''G'' by~~

Given a [[temperament mapping matrix]] ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation, represented by transformation matrix ''X'', and a ''q''-norm. Suppose the tuning is constrained by the eigenmonzo list ''M''. Let ''G'' denote the generator tuning map, we want to

~~Minimize~~

$$

\begin{align}

~~<math>~~\~~displaystyle~~ \~~norm~~{GV_X - J_X~~}_q </math>~~

& \text{find} && G \\

& \text{that minimizes} && \lVert GV_X - J_X \rVert_q \\

subject to

& \text{subject to} && (GV - J)M = O

\end{align}

~~<math>\displaystyle~~ (GV - J)~~M_I~~ = O ~~</math>~~

$$

where (·)''X'' denotes the variable in the weight–skew transformed space, found by

~~<math>\displaystyle~~

$$

\begin{align}

V_X &= VX \\

J_X &= JX

\end{align}

~~</math>~~

$$

The problem is feasible if

# rank(''M''~~''I''~~) ≤ rank(''V''), and

# rank(''M'') ≤ rank(''V''), and

# The subgroups of ''M''~~''I''~~ and N (''V'') are {{w|linear independence|linearly independent}}.

# The subgroups of ''M'' and N (''V'') are {{w|linear independence|linearly independent}}.

== Computation ==

@@ Line 11: / Line 11: @@
 == Definition ==
-Given a temperament [[mapping]] ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation, represented by transformation matrix ''X'', and a ''q''-norm. Suppose the tuning is constrained by the eigenmonzo list ''M''<sub>''I''</sub>. The goal is to find the generator list ''G'' by
+Given a [[temperament mapping matrix]] ''V'' and the [[just tuning map]] ''J'', we specify a weight–skew transformation, represented by transformation matrix ''X'', and a ''q''-norm. Suppose the tuning is constrained by the eigenmonzo list ''M''. Let ''G'' denote the generator tuning map, we want to
-Minimize
+$$
+\begin{align}
-<math>\displaystyle \norm{GV_X - J_X}_q </math>
+& \text{find} && G \\
+& \text{that minimizes} && \lVert GV_X - J_X \rVert_q \\
-subject to
+& \text{subject to} && (GV - J)M = O
+\end{align}
-<math>\displaystyle (GV - J)M_I = O </math>
+$$
 where (·)<sub>''X''</sub> denotes the variable in the weight–skew transformed space, found by
-<math>\displaystyle
+$$
 \begin{align}
 V_X &= VX \\
 J_X &= JX
 \end{align}
-</math>
+$$
 The problem is feasible if
-# rank(''M''<sub>''I''</sub>) ≤ rank(''V''), and
+# rank(''M'') ≤ rank(''V''), and
-# The subgroups of ''M''<sub>''I''</sub> and N (''V'') are {{w|linear independence|linearly independent}}.
+# The subgroups of ''M'' and N (''V'') are {{w|linear independence|linearly independent}}.
 == Computation ==